Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

2.2.10* Liftings

Suppose that and are -algebras and that is a surjective homomorphism. Given an element , an element is called a lift of if . The set of all lifts of is then the coset . We shall here be concerned with the possibility of lifting an element in with a certain property to an element in with the same property. Along this line we have the following results.

  1. Every element has a lift to an element with .
  2. Every self-adjoint element lifts to a self-adjoint element in . Moreover the self-adjoint lift can be chosen to have the same norm as .
  3. Every positive element lifts to a positive element , moreover, the positive element can be chosen so that it has the same norm as .
  4. A normal element in does not in general lift to a normal element in .
  5. A projection in does not in general lift to a projection in .
  6. Unitaries also do not lift in general to unitaries.

Proof of 2 (lifting self-adjoints):
Let be any lift of , set then is self-adjoint and . (why? lets see the full calculation)

to arrange that the lift has the same norm as , let be the continuous function given by Put Then is a normal element, being a continuous function of a normal element (Not exactly clear how we get a normal element, maybe functional calculus, but for this I thought we needed a self-adjoint element, well above we see that the definition of a_0, forces it to be self-adjoint, which in this case means we get to apply the functional calculus, the normality comes from where then? It is easy to check that the functional calculus preserves normality), and This shows that is self-adjoint, and that because the norm of is the spectral radius. Also, because for all As is a star-homomorphism, , and we conclude that .

Proof of 1 (lifting to elements with same norm):
Let be an element in , and put Then is a self-adjoint element in , and Consult exercise 1.15 regarding the third equality sign. It follows from 2. that there exists a self-adjoint lift of with . The element in is a lift of .
By (1.4) as in the proof of 2., necessarily thus

Proof of 3 (positive elements do lift):
Let be any lift of , and set then and As in the proof of 2., put Then is normal, , and Hence is positive and .

Proof of 4
See exercise 9.4(2)
Proof of 5 (Projections do not lift):
Let , let , let and let , then is a projection in , and there is no projection in such that .
Proof of 6 (Unitaries don’t lift in general):
Exercise.

Table Of Contents
2.2.10* Liftings